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MEN  OF MATHEMATICS
describe my personal scientific attitude more exactly, I may
conveniently designate it as theoretical ... ; I have particularly
striven for that mathematical knowledge which finds its proper
sphere in mathematics without reference to applications,'
Kummer was no narrow specialist. Somewhat like Gauss, he
appeared to take equal pleasure in both pure and applied
science. Gauss indeed, through his works, was Rummer*s real
teacher, and the apt pupil proved his mettle by extending his
master's work on the hypergeometric series, adding to what
Gauss had done substantial developments which to-day are of
great use in the theory of those differential equations which
recur most frequently in mathematical physics.
Again, the magnificent work of Hamilton on systems of rays
(in optics) inspired Kummer to one of his own most beautiful
inventions, that of the surface of the fourth degree which is
known by his name and which plays a fundamental part in the
geometry of Euclidean space when that space is four-dimen-
sional (instead of three-dimensionals as we ordinarily imagine
it), as happens when straight lines instead of points are taken
as the irreducible elements out of which the space is con-
structed. This surface (and its generalizations to higher spaces)
occupied the centre of the stage in a whole department of nine-
teenth-century geometry; it was found (by 'Cayley) to be repre-
sentable (parametrically - see the chapter on Gauss) by means
of the quadruple periodic functions to which Jacobi and Her-
mite devoted some of their best efforts.
Quite recently (since 1934) it has been observed by Sir Arthur
Eddington that Kummer's surface is a sort of cousin to Dirac's
wave equation in quantum mechanics (both have the same
finite group; Kummer's surface is the wave surface in space of
four dimensions)*
To complete the circle, Kummer was led back by his study of
systems of rays to physics, and he made important contribu-
tions to the theory of atmospheric refraction. In his work at the
War College he astonished the scientific world by proving him-
self a first-rate experimenter in his work on ballistics. With
characteristic humour Kummer excused himself for this bad
fall from mathematical grace: *When I attack a problem experi-
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